Local explosions and extinction in continuous-state branching processes with logistic competition

TL;DR

This paper investigates the extinction and explosion times of continuous-state branching processes with logistic competition using duality methods, introducing a bidual process to relate these phenomena to diffusion processes.

Contribution

It introduces a novel duality framework involving a bidual process to analyze extinction and explosion in LCSBPs, linking these to diffusion boundary behaviors.

Findings

Established duality relations between LCSBPs and diffusion processes.

Characterized local time at infinity for reflected processes.

Connected extinction and explosion events through dual processes.

Abstract

We study by duality methods the extinction and explosion times of continuous-state branching processes with logistic competition (LCSBPs) and identify the local time at $\infty$ of the process when it is instantaneously reflected at $\infty$. The main idea is to introduce a certain "bidual" process $V$ of the LCSBP $Z$. The latter is the Siegmund dual process of the process $U$, that was introduced in Foucart (2019), as the Laplace dual of $Z$. By using both dualities, we shall relate local explosions and the extinction of $Z$ to local extinctions and the explosion of the process $V$. The process $V$ being a one-dimensional diffusion on $[0,\infty]$, many results on diffusions can be used and transfered to $Z$. A concise study of Siegmund duality for one-dimensional diffusions and their boundaries is also provided.